How to bend one side of a planar mesh fade the bending effect towards the other side
Here is how to do it. Doing it with modifiers isn't possible as far as I know. If you need more control, comment.


Start by adding a circle. 
rotate the circle by 90 degrees on the X axis(RX90). 
Delete the bottom half of it.
Take the bottom to vertices and duplicate the by 2 units on the y axis (Shift-DY2). 

fill the two vertices.
hit (Ctrl-R).
add 1 less loop cuts then half the number of vertices in your circle you can use the scroll wheel to do this.
Join up all the faces (Ctrl-E>Bridge Edgeloops).
add some loop cuts until your faces are about square.
tweak the shape using proportional editing.